Symmetric Structures in Mock-Lie Algebras: The Quasi-Centroid and Its Matrix Representations up to Dimension 5

Abstract

Symmetric structures are key in non-associative algebras. A Mock-Lie algebra, defined by commutativity and the Jacobi identity, shows strong algebraic symmetry. This paper studies the quasi-centroid, which captures the symmetry of linear operators commuting with the algebra’s product. We define the quasi-centroid and set its condition for linear endomorphisms under the bracket operation. We classify matrix representations of quasi-centroids for all Mock-Lie algebras of dimensions 2 to 5 by computing matrices and analyzing coefficient relations. These results provide a foundation for further structural study. We also show that in each case, the centroid is strictly contained in the quasi-centroid, confirming proper containment for all these algebras.

1. Introduction

Mock-Lie algebras, also known as Jacobi–Jordan algebras, are commutative algebras that satisfy the Jacobi identity [1,2]. They occupy a unique position between Jordan algebras and Lie algebras, sharing features of both while remaining distinct. The study of Mock-Lie algebras began with early work on commutative algebras [3]. Interest in them has grown recently, especially for symmetric structures and their applications in mathematical physics [4,5,6]. A systematic study in [7] focused on infinite-dimensional and solvable but non-nilpotent algebras. Additional properties and examples are established in [8,9]. The classification of low-dimensional Mock-Lie algebras was advanced in [1], which provides a complete classification up to dimension 7 over algebraically closed fields of characteristic not 2 or 3. Recent developments cover Rota–Baxter structures, Yang–Baxter equations, and compatible bialgebras related to Mock-Lie algebras [4,5,10].

The quasi-centroid, or commuting linear maps [11,12,13,14], is a key structure for understanding many algebras [15]. For a Mock-Lie algebra L, it is the set of linear transformations $\phi$ satisfying $\left[\phi \right(x),y]=[x,\phi (y\left)\right]$ for all $x,y\in L$. This definition reveals important features of the algebra and its symmetry. Quasi-centroids are closely related to biderivations of Lie algebras [13,14] due to similarities in their defining conditions. Recent studies detail matrix representations for centroids in low-dimensional Mock-Lie algebras [16] and examine special cases [17]. Building on these foundations, quasi-centroids also play roles in applied mathematics, including molecular dynamics [18] and fractional calculus [19]. In molecular dynamics, for example, they enable symmetric transformations, which help study symmetric reaction pathways and molecular transition states. Our matrix representations assist in finding symmetry-preserving transformations in complex systems, since the quasi-centroid condition $\left[\phi \right(x),y]=[x,\phi (y\left)\right]$ detects linear maps that preserve algebra symmetries. Structurally, the quasi-centroid captures a distinct kind of linear symmetry by imposing symmetric constraints on a linear map’s interaction with the algebraic product, making it a natural tool for probing symmetries in non-associative algebras.

Although centroids and related structures have been investigated in Mock-Lie algebras [20,21,22], matrix representations of quasi-centroids in low dimensions remain unexplored. These representations have potential applications in mathematical physics [23] and in the analysis of algebraic symmetries, particularly in cases where practical implementations require explicit computational forms. We aim to fill this gap by obtaining complete matrix representations of quasi-centroids of Mock-Lie algebras of dimension up to 5. Focusing on dimensions $n\le 5$ provides a foundational step for several reasons: the classification of these algebras is complete and well-established [1], allowing for a systematic computational exploration; furthermore, the structural patterns and relationships uncovered in low dimensions often provide crucial insights and conjectures for understanding the behavior of these algebras in higher dimensions or more general settings. We extend the classification of [1] to quasi-centroids, using previous work on centroids [16], to develop concrete computational tools for analyzing these structures. We find that, for all Mock-Lie algebras of dimension up to 5, the centroid is strictly contained within the quasi-centroid. We establish the first comprehensive framework for studying quasi-centroid structures in low-dimensional Mock-Lie algebras. It may be used for biderivations and symmetric transformations, as well as for molecular dynamics simulations involving symmetric algebraic structures.

This paper is organized as follows: Section 2 reviews key concepts and definitions, including the formal definitions of Mock-Lie algebras and quasi-centroids. Section 3 then presents our main results—explicit matrix representations of quasi-centroids for Mock-Lie algebras of dimensions 2 to 5. Finally, Section 4 offers a comparative analysis of centroids and quasi-centroids, discusses the significance of our results, proposes applications in mathematical physics, and outlines directions for future research.

2. Preliminaries

Throughout this paper, we assume that the base field $\mathbb{K}$ has characteristics different from 2 and 3. All algebras considered are finite-dimensional over $\mathbb{K}$.

Definition 1 

([1,17]). A Mock-Lie algebrais a vector space L over a field $\mathbb{K}$, equipped with a bilinear map $[·,·]:L\times L\to L$, called the bracket, satisfying the following axioms for all $x,y,z\in L$:

• Symmetry: $[x,y]=[y,x]$;
• Jacobi Identity: $\left[[x,y],z\right]+\left[[z,x],y\right]+\left[[y,z],x\right]=0$.

Mock-Lie algebras thus combine the symmetry of Jordan algebras with the Jacobi identity characteristic of Lie algebras, forming an interesting class of non-associative algebras.

For computational purposes, we fix a basis $\{e_1,\dots ,e_n\}$ of L. The bracket product is completely determined by its values on basis elements, expressed via the structure constants $c_{ij}^k\in \mathbb{K}$ defined by

$$[e_i,e_j]=\sum _{k=1}^n{c}_{ij}^k{e}_k.$$

Definition 2 

([15]). Let L be a Mock-Lie algebra over $\mathbb{K}$. The quasi-centroid of L is defined as the set of linear transformations:

$$QC\left(L\right)=\{\phi :L\to L\mid [\phi \left(x\right),y]=[x,\phi \left(y\right)]\mathrm{for}\mathrm{all}x,y\in L\}.$$

The quasi-centroid comprises linear maps that commute with the bracket operation. It generalizes the classical centroid while preserving key algebraic properties. The defining equation $\left[\phi \right(x),y]=[x,\phi (y\left)\right]$ is inherently a symmetry condition. The output remains unchanged regardless of which argument the linear map $\phi$ acts upon, thanks to the commutativity of the product.

We now state a lemma, analogous to the one in the Lie algebra case [20], that provides a practical test for identifying elements of the quasi-centroid. This lemma offers an explicit, computational criterion, making it easier to verify whether a linear map belongs to the quasi-centroid in concrete examples.

Lemma 1. 

Let L be a finite-dimensional Mock-Lie algebra with basis $\{e_1,\dots ,e_n\}$. A linear transformation $\phi :L\to L$ belongs to $QC\left(L\right)$ if and only if

$$[\phi \left(e_i\right),e_j]=[e_i,\phi \left(e_j\right)]foralli,j\in \{1,2,\dots ,n\}.$$

Proof. 

Necessity follows directly from Definition 2. For sufficiency, consider arbitrary elements $x={\sum }_{i=1}^n{\alpha }_i{e}_i$ and $y={\sum }_{j=1}^n{\beta }_j{e}_j$ in L. Assuming the condition holds on basis elements, we compute the following:

$$\begin{array}{cc}\hfill \left[\phi \right(x),y]& =\left[\phi \left(\sum _{i=1}^n{\alpha }_i{e}_i\right),\sum _{j=1}^n{\beta }_j{e}_j\right]=\left[\sum _{i=1}^n{\alpha }_i\phi \left(e_i\right),\sum _{j=1}^n{\beta }_j{e}_j\right]\hfill \\ \hfill & =\sum _{i=1}^n\sum _{j=1}^n{\alpha }_i{\beta }_j[\phi \left(e_i\right),e_j]=\sum _{i=1}^n\sum _{j=1}^n{\alpha }_i{\beta }_j[e_i,\phi \left(e_j\right)]\hfill \\ \hfill & =\left[\sum _{i=1}^n{\alpha }_i{e}_i,\sum _{j=1}^n{\beta }_j\phi \left(e_j\right)\right]=\left[x,\phi \left(\sum _{j=1}^n{\beta }_j{e}_j\right)\right]=[x,\phi \left(y\right)].\hfill \end{array}$$

This establishes the sufficiency condition. □

For computational purposes, we fix a basis $\{e_1,\dots ,e_n\}$ of L and represent linear transformations using matrices. If $\phi :L\to L$ has a matrix representation $A=\left(a_{ij}\right)$ relative to this basis, meaning

$$\phi \left(e_j\right)=\sum _{i=1}^n{a}_{ij}{e}_i\mathrm{for}j=1,\dots ,n.$$

Then Lemma 1 allows us to translate the quasi-centroid condition into a system of equations relating the matrix entries $a_{ij}$ and the structure constants of the Mock-Lie algebra.

This paper determines the explicit constraints on the matrix A imposed by the quasi-centroid condition for all low-dimensional (≤5) Mock-Lie algebras, building upon their known classification. For the reader’s convenience, we summarize the main notations used throughout this paper in Abbreviations.

3. Main Results and Proof

Computation Procedure: The explicit matrix representations in this section were derived through the following systematic procedure: For each algebra, the quasi-centroid condition from Lemma 1 was applied, generating a system of linear equations in the matrix entries $a_{ij}$. The generation and initial simplification of this large system of equations were performed using symbolic computation software (Wolfram Mathematica 14.2) to ensure accuracy and efficiency. Subsequently, the final solving of the system, analysis of the solution space, and construction of the basis matrices (as listed in Theorem 1–4) were carried out manually. This hybrid approach ensures computational rigor while providing clear, interpretable results.

Proposition 1. 

Every Mock-Lie algebra of dimension $n\le 5$ is isomorphic to one of the algebras listed in Table 1.

Table 1. Classification of Mock-Lie algebras with dimension $n\le 5$.

Algebras	Multiplication Rules	Dimension
$A_{0,1}$	–	1
$A_{0,1}\oplus A_{0,1}$	–	2
$A_{1,2}$	$[e_1,e_1]=e_2$	2
$A_{0,1}\oplus A_{0,1}\oplus A_{0,1}$	–	3
$A_{1,2}\oplus A_{0,1}$	$[e_1,e_1]=e_2$	3
$A_{1,3}$	$[e_1,e_1]=e_2$, $[e_3,e_3]=e_2$	3
$A_{0,1}^4$	–	4
$A_{1,2}\oplus A_{0,1}^2$	$[e_1,e_1]=e_2$	4
$A_{1,3}\oplus A_{0,1}$	$[e_1,e_1]=e_2$, $[e_3,e_3]=e_2$	4
$A_{1,2}\oplus A_{1,2}$	$[e_1,e_1]=e_2$, $[e_3,e_3]=e_4$	4
$A_{1,4}$	$[e_1,e_1]=e_2$, $[e_1,e_3]=[e_3,e_1]=e_4$	4
$A_{2,4}$	$[e_1,e_1]=e_2$, $[e_3,e_4]=[e_4,e_3]=e_2$	4
$A_{1,5}$	$[e_1,e_1]=e_2$, $[e_1,e_3]=[e_3,e_1]=e_5$, $[e_3,e_3]=e_4$	5
$A_{2,5}$	$[e_1,e_1]=e_2$, $[e_1,e_4]=[e_4,e_1]=e_5$, $[e_3,e_3]=e_5$	5
$A_{3,5}$	$[e_1,e_1]=e_2$, $[e_1,e_4]=[e_4,e_1]=e_5$, $[e_3,e_3]=-e_2+e_5$, $[e_3,e_4]=[e_4,e_3]=e_5$	5
$A_{4,5}$	$[e_1,e_1]=e_2$, $[e_3,e_3]=e_4$, $[e_5,e_5]=-e_2+e_4$	5
$A_{5,5}$	$[e_1,e_1]=e_2$, $[e_3,e_3]=e_4$, $[e_3,e_5]=[e_5,e_3]=-e_2+e_4$	5
$A_{6,5}$	$[e_1,e_1]=e_2$, $[e_1,e_3]=[e_3,e_1]=e_5$, $[e_1,e_4]=[e_4,e_1]={\displaystyle \frac{1}{2}}{e}_2$	5
$A_{7,5}$	$[e_1,e_1]=e_2$, $[e_3,e_3]=-e_2$, $[e_4,e_5]=[e_5,e_4]={\displaystyle \frac{1}{2}}{e}_2$	5
$A_{0,1}^5$	–	5
$A_{1,2}\oplus A_{0,1}^3$	$[e_1,e_1]=e_2$	5
$A_{1,3}\oplus A_{0,1}^2$	$[e_1,e_1]=e_2$, $[e_3,e_3]=e_2$	5
$A_{2,4}\oplus A_{0,1}$	$[e_1,e_1]=e_2$, $[e_3,e_4]=[e_4,e_3]=e_2$	5
$A_{1,2}^2\oplus A_{0,1}$	$[e_1,e_1]=e_2$, $[e_3,e_3]=e_4$	5
$A_{1,2}\oplus A_{1,3}$	$[e_1,e_1]=e_2$, $[e_3,e_3]=e_4$, $[e_5,e_5]=e_4$	5
$A_{1,4}\oplus A_{0,1}$	$[e_1,e_1]=e_2$, $[e_1,e_3]=[e_3,e_1]=e_4$	5

Table 1 shows that algebras $A_{0,1}$, $A_{0,1}\oplus A_{0,1}$, $A_{0,1}\oplus A_{0,1}\oplus A_{0,1}$, $A_{0,1}^4$, and $A_{0,1}^5$ are abelian; i.e., all products vanish. Since the quasi-centroid of an abelian algebra coincides with the space of all linear endomorphisms, we focus our analysis exclusively on the non-abelian cases.

We now present our main results, beginning with the two-dimensional case.

Theorem 1. 

The matrix representation of the quasi-centroid for the two-dimensional non-abelian Mock-Lie algebra $A_{1,2}$ with respect to the basis $\{e_1,e_2\}$ is given by

$$QC\left(A_{1,2}\right)=\left\{\left(\begin{array}{cc}{a}_{11}& 0\\ a_{21}& a_{22}\end{array}\right):a_{11},a_{21},a_{22}\in \mathbb{K}\right\}.$$

A basis for $QC\left(A_{1,2}\right)$ is provided in Table 2.

Table 2. Basis for $QC\left(L\right)$ of the two-dimensional non-abelian Mock-Lie algebra.

Algebra	Basis for $\mathit{Q}\mathit{C}\mathbf{\left(}\mathit{L}\mathbf{\right)}$
$A_{1,2}$	$E_{11},E_{21},E_{22}$

Proof. 

Consider the algebra $A_{1,2}$ with the only non-zero bracket $[e_1,e_1]=e_2$. Let $\phi :L\to L$ be a linear transformation with matrix representation

$$A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)$$

relative to the basis $\{e_1,e_2\}$, so that

$$\phi \left(e_1\right)=a_{11}{e}_1+a_{21}{e}_2\mathrm{and}\phi \left(e_2\right)=a_{12}{e}_1+a_{22}{e}_2.$$

Using Lemma 1, $\phi \in QC\left(L\right)$ if and only if the following conditions hold for all basis elements:

$$\begin{array}{cc}\hfill [\phi \left(e_1\right),e_1]& =[e_1,\phi \left(e_1\right)],\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill [\phi \left(e_1\right),e_2]& =[e_1,\phi \left(e_2\right)],\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill [\phi \left(e_2\right),e_1]& =[e_2,\phi \left(e_1\right)],\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill [\phi \left(e_2\right),e_2]& =[e_2,\phi \left(e_2\right)].\hfill \end{array}$$

(4)

Condition (1) holds identically because the bracket is commutative. Similarly, condition (4) also holds trivially since all brackets involving $e_2$ are zero. Therefore, we focus on conditions (2) and (3). Since

$$\begin{array}{cc}[\phi \left(e_1\right),e_2]\hfill & =[a_{11}{e}_1+a_{21}{e}_2,e_2]=a_{11}[e_1,e_2]+a_{21}[e_2,e_2]=0,\hfill \\ [e_1,\phi \left(e_2\right)]\hfill & =[e_1,a_{12}{e}_1+a_{22}{e}_2]=a_{12}[e_1,e_1]+a_{22}[e_1,e_2]=a_{12}{e}_2,\hfill \end{array}$$

condition (2) implies $a_{12}=0$. It follows from

$$\begin{array}{cc}[\phi \left(e_2\right),e_1]\hfill & =[a_{12}{e}_1+a_{22}{e}_2,e_1]=a_{12}[e_1,e_1]+a_{22}[e_2,e_1]=a_{12}{e}_2,\hfill \\ [e_2,\phi \left(e_1\right)]\hfill & =[e_2,a_{11}{e}_1+a_{21}{e}_2]=a_{11}[e_2,e_1]+a_{21}[e_2,e_2]=0.\hfill \end{array}$$

that condition (3) is automatically satisfied. The only nontrivial coefficient relation is $a_{12}=0$. Therefore, the matrix representation of $\phi \in QC\left(A_{1,2}\right)$ is

$$A=\left(\begin{array}{cc}{a}_{11}& 0\\ a_{21}& a_{22}\end{array}\right),$$

which corresponds to the basis $\{E_{11},E_{21},E_{22}\}$ as claimed. □

Theorem 2. 

The matrix representations of quasi-centroids for three-dimensional Mock-Lie algebras, with respect to their standard bases, are given by the bases listed in Table 3.

Table 3. Bases for $QC\left(L\right)$ of the three-dimensional non-abelian Mock-Lie algebras.

Algebra	Basis for $\mathit{Q}\mathit{C}\mathbf{\left(}\mathit{L}\mathbf{\right)}$
$A_{1,2}\oplus A_{0,1}$	$E_{11},E_{21},E_{22},E_{23},E_{31},E_{32},E_{33}$
$A_{1,3}$	$E_{11},E_{13}+E_{31},E_{21},E_{22},E_{23},E_{33}$

Proof. 

Applying Lemma 1 to determine the constraints on the matrix entries, we analyze each algebra separately.

Case 1: Algebra $A_{1,2}\oplus A_{0,1}$.

This algebra has multiplication $[e_1,e_1]=e_2$ with all other brackets equal to zero. Let $\phi :L\to L$ have matrix representation

$$A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$$

relative to the basis $\{e_1,e_2,e_3\}$.

The quasi-centroid condition requires $[\phi \left(e_i\right),e_j]=[e_i,\phi \left(e_j\right)]$ for all $i,j\in \{1,2,3\}$. Due to commutativity symmetry, many conditions are automatically satisfied. The nontrivial constraints are listed below:

• For $i=1$ and $j=2$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_2]\hfill & =[a_{11}{e}_1+a_{21}{e}_2+a_{31}{e}_3,e_2]=0,\hfill \\ [e_1,\phi \left(e_2\right)]\hfill & =[e_1,a_{12}{e}_1+a_{22}{e}_2+a_{32}{e}_3]=a_{12}[e_1,e_1]=a_{12}{e}_2.\hfill \end{array}$$
  Thus $a_{12}=0$.
• For $i=1$ and $j=3$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_3]\hfill & =[a_{11}{e}_1+a_{21}{e}_2+a_{31}{e}_3,e_3]=0,\hfill \\ [e_1,\phi \left(e_3\right)]\hfill & =[e_1,a_{13}{e}_1+a_{23}{e}_2+a_{33}{e}_3]=a_{13}[e_1,e_1]=a_{13}{e}_2.\hfill \end{array}$$
  Thus $a_{13}=0$.

Therefore, the matrix representation of $\phi \in QC(A_{1,2}\oplus A_{0,1})$ is

$$A=\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right),$$

which corresponds to the basis $\{E_{11},E_{21},E_{22},E_{23},E_{31},E_{32},E_{33}\}$.

Case 2: Algebra $A_{1,3}$.

This algebra has multiplications $[e_1,e_1]=e_2$ and $[e_3,e_3]=e_2$, with all other products equal to zero. Using the same matrix representation $A=\left(a_{ij}\right)$, the nontrivial constraints are listed below:

• For $i=1$ and $j=2$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_2]\hfill & =[a_{11}{e}_1+a_{21}{e}_2+a_{31}{e}_3,e_2]=0,\hfill \\ [e_1,\phi \left(e_2\right)]\hfill & =[e_1,a_{12}{e}_1+a_{22}{e}_2+a_{32}{e}_3]=a_{12}[e_1,e_1]=a_{12}{e}_2.\hfill \end{array}$$
  Thus $a_{12}=0$.
• For $i=1$ and $j=3$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_3]\hfill & =[a_{11}{e}_1+a_{21}{e}_2+a_{31}{e}_3,e_3]=a_{31}[e_3,e_3]=a_{31}{e}_2,\hfill \\ [e_1,\phi \left(e_3\right)]\hfill & =[e_1,a_{13}{e}_1+a_{23}{e}_2+a_{33}{e}_3]=a_{13}[e_1,e_1]=a_{13}{e}_2.\hfill \end{array}$$
  Thus $a_{31}=a_{13}$.
• For $i=3$ and $j=2$,

  $$\begin{array}{cc}[\phi \left(e_3\right),e_2]\hfill & =[a_{13}{e}_1+a_{23}{e}_2+a_{33}{e}_3,e_2]=0,\hfill \\ [e_3,\phi \left(e_2\right)]\hfill & =[e_3,a_{12}{e}_1+a_{22}{e}_2+a_{32}{e}_3]=a_{32}[e_3,e_3]=a_{32}{e}_2.\hfill \end{array}$$
  Thus $a_{32}=0$.
• For $i=3$ and $j=1$,

  $$\begin{array}{cc}[\phi \left(e_3\right),e_1]\hfill & =[a_{13}{e}_1+a_{23}{e}_2+a_{33}{e}_3,e_1]=a_{13}[e_1,e_1]=a_{13}{e}_2,\hfill \\ [e_3,\phi \left(e_1\right)]\hfill & =[e_3,a_{11}{e}_1+a_{21}{e}_2+a_{31}{e}_3]=a_{31}[e_3,e_3]=a_{31}{e}_2.\hfill \end{array}$$
  Thus $a_{31}=a_{13}$.

The nontrivial coefficient relations are $a_{12}=a_{32}=0$ and $a_{13}=a_{31}$. Therefore, the matrix representation of $\phi \in QC\left(A_{1,3}\right)$ is

$$A=\left(\begin{array}{ccc}{a}_{11}& 0& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{13}& 0& a_{33}\end{array}\right),$$

which corresponds to the basis $\{E_{11},E_{13}+E_{31},E_{21},E_{22},E_{23},E_{33}\}$.

This completes the proof for both three-dimensional non-abelian Mock-Lie algebras. □

Theorem 3. 

The matrix representations of quasi-centroids for four-dimensional Mock-Lie algebras, with respect to their standard bases, are given by the bases listed in Table 4.

Table 4. Bases for $QC\left(L\right)$ of the four-dimensional non-abelian Mock-Lie algebras.

Algebra	Basis for $\mathit{Q}\mathit{C}\mathbf{\left(}\mathit{L}\mathbf{\right)}$
$A_{1,2}\oplus A_{0,1}^2$	$E_{11},E_{21},E_{22},E_{23},E_{24},E_{31},E_{32},E_{33},E_{34},E_{41},E_{42},E_{43},E_{44}$
$A_{1,3}\oplus A_{0,1}$	$E_{11},E_{13}+E_{31},E_{33},E_{21},E_{22},E_{23},E_{24},E_{41},E_{42},E_{43},E_{44}$
$A_{1,2}\oplus A_{1,2}$	$E_{11},E_{33},E_{21},E_{22},E_{23},E_{24},E_{41},E_{42},E_{43},E_{44}$
$A_{1,4}$	$E_{11}+E_{33},E_{31},E_{21},E_{22},E_{23},E_{24},E_{41},E_{42},E_{43},E_{44}$
$A_{2,4}$	$E_{11},E_{21},E_{22},E_{23},E_{24},E_{31}+E_{14},E_{13}+E_{41},E_{33}+E_{44},E_{34},E_{43}$

Proof. 

Applying Lemma 1, we analyze each four-dimensional non-abelian Mock-Lie algebra separately. For each algebra, we consider a linear transformation $\phi :L\to L$ with matrix representation $A={(a_{ij})}_{4\times 4}$ relative to the basis $\{e_1,e_2,e_3,e_4\}$.

Case 1: Algebra $A_{1,2}\oplus A_{0,1}^2$.

This algebra has multiplication $[e_1,e_1]=e_2$ with all other brackets equal to zero. The nontrivial constraints are listed below:

• For $[\phi \left(e_1\right),e_i]=[e_1,\phi \left(e_i\right)]$ with $i=2,3,4$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_i]\hfill & =[a_{11}{e}_1+a_{21}{e}_2+a_{31}{e}_3+a_{41}{e}_4,e_i]=0,\hfill \\ [e_1,\phi \left(e_i\right)]\hfill & =[e_1,a_{1i}{e}_1+a_{2i}{e}_2+a_{3i}{e}_3+a_{4i}{e}_4]=a_{1i}[e_1,e_1]=a_{1i}{e}_2.\hfill \end{array}$$
  Thus, $a_{1i}=0$ for $i=2,3,4$.

Therefore, the matrix representation is

$$A=\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{31}& a_{32}& a_{33}& a_{34}\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right),$$

which corresponds to the basis in Table 4.

Case 2: Algebra $A_{1,3}\oplus A_{0,1}$.

This algebra has multiplications $[e_1,e_1]=e_2$ and $[e_3,e_3]=e_2$, with all other products equal to zero. The nontrivial constraints are listed below.

• For $[\phi \left(e_1\right),e_i]=[e_1,\phi \left(e_i\right)]$ with $i=2,3,4$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_2]\hfill & =0=[e_1,\phi \left(e_2\right)]=a_{12}{e}_2,\hfill \\ [\phi \left(e_1\right),e_3]\hfill & =a_{31}[e_3,e_3]=a_{31}{e}_2=[e_1,\phi \left(e_3\right)]=a_{13}{e}_2,\hfill \\ [\phi \left(e_1\right),e_4]\hfill & =0=[e_1,\phi \left(e_4\right)]=a_{14}{e}_2.\hfill \end{array}$$
  Then $a_{12}=a_{14}=0$ and $a_{31}=a_{13}$.
• For $[\phi \left(e_3\right),e_i]=[e_3,\phi \left(e_i\right)]$ with $i=2,4$,

  $$\begin{array}{cc}[\phi \left(e_3\right),e_2]\hfill & =0=[e_3,\phi \left(e_2\right)]=a_{32}{e}_2,\hfill \\ [\phi \left(e_3\right),e_4]\hfill & =0=[e_3,\phi \left(e_4\right)]=a_{34}{e}_2.\hfill \end{array}$$
  Then $a_{32}=a_{34}=0$.

Thus, nontrivial coefficient relations are $a_{12}=a_{14}=a_{32}=a_{34}=0$ and $a_{31}=a_{13}$. The matrix representation is

$$A=\left(\begin{array}{cccc}{a}_{11}& 0& a_{13}& 0\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{13}& 0& a_{33}& 0\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right),$$

corresponding to the basis in Table 4.

Case 3: Algebra $A_{1,2}\oplus A_{1,2}$.

This algebra has multiplications $[e_1,e_1]=e_2$ and $[e_3,e_3]=e_4$, with all other products equal to zero. The analysis yields that nontrivial coefficient relations are $a_{1i}=0$ for $i=2,3,4$ and $a_{3j}=0$ for $j=1,2,4$. The matrix representation is

$$A=\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}\\ 0& 0& a_{33}& 0\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right),$$

corresponding to the basis in Table 4.

Case 4: Algebra $A_{1,4}$.

This algebra has multiplications $[e_1,e_1]=e_2$ and $[e_1,e_3]=[e_3,e_1]=e_4$, with all other products equal to zero. The nontrivial constraints are listed below.

• For $[\phi \left(e_1\right),e_i]=[e_1,\phi \left(e_i\right)]$ with $i=2,3,4$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_2]\hfill & =0=[e_1,\phi \left(e_2\right)]=a_{12}{e}_2+a_{32}{e}_4,\hfill \\ [\phi \left(e_1\right),e_3]\hfill & =a_{11}{e}_4=[e_1,\phi \left(e_3\right)]=a_{13}{e}_2+a_{33}{e}_4,\hfill \\ [\phi \left(e_1\right),e_4]\hfill & =0=[e_1,\phi \left(e_4\right)]=a_{14}{e}_2+a_{34}{e}_4.\hfill \end{array}$$
  From these, we obtain $a_{11}=a_{33}$, and $a_{12}=a_{13}=a_{14}=a_{32}=a_{34}=0$.
• For $[\phi \left(e_4\right),e_1]=[e_4,\phi \left(e_1\right)]$,

  $$\begin{array}{cc}[\phi \left(e_4\right),e_1]\hfill & =a_{14}[e_1,e_1]+a_{34}[e_3,e_1]=a_{14}{e}_2+a_{34}{e}_4,\hfill \\ [e_4,\phi \left(e_1\right)]\hfill & =0.\hfill \end{array}$$
  This yields $a_{14}=0$ and $a_{34}=0$.

Combining all constraints gives $a_{12}=a_{13}=a_{14}=a_{32}=a_{34}=0$, and $a_{11}=a_{33}$. The matrix representation is

$$A=\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{31}& 0& a_{11}& 0\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right),$$

corresponding to the basis in Table 4.

Case 5: Algebra $A_{2,4}$.

This algebra has multiplications $[e_1,e_1]=e_2$ and $[e_3,e_4]=[e_4,e_3]=e_2$, with all other products equal to zero. The nontrivial constraints are listed below:

• For $[\phi \left(e_1\right),e_i]=[e_1,\phi \left(e_i\right)]$ with $i=2,3,4$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_2]\hfill & =0=[e_1,\phi \left(e_2\right)]=a_{12}{e}_2,\hfill \\ [\phi \left(e_1\right),e_3]\hfill & =a_{31}[e_3,e_1]+a_{41}[e_4,e_1]=0=[e_1,\phi \left(e_3\right)]=a_{13}{e}_2,\hfill \\ [\phi \left(e_1\right),e_4]\hfill & =a_{31}[e_3,e_4]+a_{41}[e_4,e_4]=a_{31}{e}_2=[e_1,\phi \left(e_4\right)]=a_{14}{e}_2.\hfill \end{array}$$
  Then $a_{12}=a_{13}=0$ and $a_{31}=a_{14}$.
• For $[\phi \left(e_2\right),e_j]=[e_2,\phi \left(e_j\right)]$ with $j=3,4$,

  $$\begin{array}{cc}[\phi \left(e_2\right),e_3]\hfill & =a_{32}[e_3,e_3]+a_{42}[e_4,e_3]=a_{42}{e}_2=[e_2,\phi \left(e_3\right)]=0,\hfill \\ [\phi \left(e_2\right),e_4]\hfill & =a_{32}[e_3,e_4]+a_{42}[e_4,e_4]=a_{32}{e}_2=[e_2,\phi \left(e_4\right)]=0.\hfill \end{array}$$
  Then $a_{42}=a_{32}=0$.
• For $[\phi \left(e_4\right),e_3]=[e_4,\phi \left(e_3\right)]$,

  $$\begin{array}{cc}[\phi \left(e_4\right),e_3]\hfill & =a_{41}[e_1,e_3]+a_{43}[e_3,e_3]+a_{44}[e_4,e_3]=a_{44}{e}_2,\hfill \\ [e_4,\phi \left(e_3\right)]\hfill & =a_{31}[e_4,e_1]+a_{33}[e_4,e_3]=a_{33}{e}_2.\hfill \end{array}$$
  Then $a_{44}=a_{33}$.
• Additional constraints yield $a_{i2}=0$ for $i=1,3,4$ and $a_{13}=a_{41}$.

The nontrivial coefficient relations are $a_{12}=a_{13}=a_{32}=a_{42}=0$, $a_{31}=a_{14}$, $a_{13}=a_{41}$, and $a_{33}=a_{44}$. The matrix representation is

$$A=\left(\begin{array}{cccc}{a}_{11}& 0& a_{13}& a_{14}\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{14}& 0& a_{33}& a_{34}\\ a_{13}& 0& a_{43}& a_{33}\end{array}\right),$$

corresponding to the basis in Table 4.

This completes the proof for all four-dimensional non-abelian Mock-Lie algebras. □

Theorem 4. 

The matrix representations of quasi-centroids for five-dimensional Mock-Lie algebras, with respect to their standard bases, are given by the bases listed in Table 5.

Table 5. Bases for $QC\left(L\right)$ of the five-dimensional non-abelian Mock-Lie algebras.

Algebra	Basis for $\mathit{Q}\mathit{C}\mathbf{\left(}\mathit{L}\mathrm{\right)}$
$A_{1,5}$	$E_{11}+E_{33},E_{2i},E_{4i},E_{5i}$
$A_{2,5}$	$E_{11}+E_{44},E_{31}+E_{43},E_{2i},E_{33},E_{34},E_{35},E_{41},E_{5i}$
$A_{3,5}$	$E_{11}+E_{13}+E_{31}-E_{43}+E_{44},E_{33}+E_{44},E_{41}+E_{43},E_{2i},E_{5i}$
$A_{4,5}$	$E_{11},E_{33},E_{55},E_{2i},E_{4i}$
$A_{5,5}$	$E_{11},E_{33},E_{53},E_{55},E_{2i},E_{4i}$
$A_{6,5}$	$E_{11}+E_{33},E_{31},E_{41},E_{44},E_{2i},E_{5i}$
$A_{7,5}$	$E_{13}-E_{31},E_{14}+2E_{51},E_{15}+2E_{41},E_{35}-2E_{43},E_{34}-2E_{53},E_{45},$
	$E_{54},E_{44}+E_{55},E_{11},E_{33},E_{2i}$
$A_{1,2}\oplus A_{0,1}^3$	$E_{11},E_{2i},E_{3i},E_{4i},E_{5i}$
$A_{1,3}\oplus A_{0,1}^2$	$E_{13}+E_{31},E_{11},E_{33},E_{2i},E_{4i},E_{5i}$
$A_{2,4}\oplus A_{0,1}$	$E_{14}+E_{31},E_{41}+E_{13},E_{33}+E_{44},E_{11},E_{43},E_{2i},E_{5i}$
$A_{1,2}^2\oplus A_{0,1}$	$E_{13}+E_{31},E_{11},E_{33},E_{2i},E_{4i},E_{5i}$
$A_{1,2}\oplus A_{1,3}$	$E_{13}+E_{31},E_{11},E_{15},E_{33},E_{35},E_{51},E_{53},E_{55},E_{2i},E_{4i}$
$A_{1,4}\oplus A_{0,1}$	$E_{11}+E_{33},E_{31},E_{2i},E_{4i},E_{5i}$

i = 1, 2, …, 5.

Proof. 

We analyze each five-dimensional non-abelian Mock-Lie algebra systematically, applying Lemma 1. For each algebra, we consider a linear transformation $\phi :L\to L$ with matrix representation $A={\left(a_{ij}\right)}_{5\times 5}$ relative to the basis $\{e_1,e_2,e_3,e_4,e_5\}$.

Case 1: Algebra $A_{1,5}$.

This algebra has non-zero brackets

$$[e_1,e_1]=e_2,[e_1,e_3]=[e_3,e_1]=e_5,[e_3,e_3]=e_4.$$

The nontrivial constraints are listed below:

• For $[\phi \left(e_1\right),e_i]=[e_1,\phi \left(e_i\right)]$ with $i=2,3,4,5$,

  $$\begin{array}{cc}[\phi \left(e_1\right),e_2]\hfill & =0=[e_1,\phi \left(e_2\right)]=a_{12}{e}_2,\hfill \\ [\phi \left(e_1\right),e_3]\hfill & =a_{31}[e_3,e_1]+a_{33}[e_3,e_3]=-a_{31}{e}_5+a_{33}{e}_4,\hfill \\ [e_1,\phi \left(e_3\right)]\hfill & =a_{13}[e_1,e_1]+a_{33}[e_1,e_3]=a_{13}{e}_2+a_{33}{e}_5.\hfill \end{array}$$
  Comparing coefficients gives $a_{31}=-a_{33}$ and $a_{12}=a_{13}=0$.
• For $[\phi \left(e_2\right),e_3]=[e_2,\phi \left(e_3\right)]$,

  $$\begin{array}{cc}[\phi \left(e_2\right),e_3]\hfill & =a_{32}[e_3,e_3]=a_{32}{e}_4,\hfill \\ [e_2,\phi \left(e_3\right)]\hfill & =0.\hfill \end{array}$$
  Then $a_{32}=0$.
• Additional analysis yields $a_{1i}=0$ for $i=2,3,4,5$, $a_{3j}=0$ for $j=1,2,4,5$, and $a_{11}=a_{33}$.

The matrix representation is

$$A=\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ 0& 0& a_{11}& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right).$$

The treatment methods for the remaining types of algebras are similar. Therefore, only the non-zero brackets and the relationships between the matrix entries are listed as follows:

Case 2: Algebra $A_{2,5}$.

Non-zero brackets:

$$[e_1,e_1]=e_2,[e_1,e_4]=[e_4,e_1]=e_5,[e_3,e_3]=e_5.$$

Matrix entries:

$a_{1i}=0$ for $i=2,3,4,5$, $a_{4j}=0$ for $j=2,5$, $a_{32}=0$, $a_{31}=a_{43}$, and $a_{11}=a_{44}$.

Case 3: Algebra $A_{3,5}$.

Non-zero brackets:

$$[e_1,e_1]=e_2,[e_1,e_4]=[e_4,e_1]=e_5,[e_3,e_3]=-e_2+e_5,[e_3,e_4]=[e_4,e_3]=e_5.$$

Matrix entries:

$a_{11}=a_{13}=a_{31}$, $a_{1i}=a_{3i}=a_{42}=a_{45}=0$ for $i=2,4,5$, $a_{43}=a_{41}-a_{11}$, and $a_{44}=a_{11}+a_{33}$.

Case 4: Algebra $A_{4,5}$.

Non-zero brackets:

$$[e_1,e_1]=e_2,[e_3,e_3]=e_4,[e_5,e_5]=-e_2+e_4.$$

Matrix entries:

$a_{1i}=0$ for $i=2,3,4,5$, $a_{3j}=0$ for $j=1,2,4,5$, and $a_{5n}=0$ for $n=1,2,3,4$.

Case 5: Algebra $A_{5,5}$.

Non-zero brackets:

$$[e_1,e_1]=e_2,[e_3,e_3]=e_4,[e_3,e_5]=[e_5,e_3]=-e_2+e_4.$$

Matrix entries:

$a_{1i}=0$ for $i=2,3,4,5$, $a_{3j}=0$ for $j=1,2,4,5$, and $a_{5n}=0$ for $n=1,2,4$.

Case 6: Algebra $A_{6,5}$.

Non-zero brackets:

$$[e_1,e_1]=e_2,[e_1,e_3]=[e_3,e_1]=e_5,[e_1,e_4]=[e_4,e_1]=\frac{1}{2}{e}_2.$$

Matrix entries:

$a_{11}=a_{33}$, $a_{1i}=0$ for $i=2,3,4,5$, $a_{3j}=0$ for $j=2,4,5$, and $a_{4n}=0$ for $n=2,3,5$.

Case 7: Algebra $A_{7,5}$.

Non-zero brackets:

$$[e_1,e_1]=e_2,[e_3,e_3]=-e_2,[e_4,e_5]=[e_5,e_4]=\frac{1}{2}{e}_2.$$

Matrix entries:

$a_{i2}=0$ for $i=1,3,4,5$, $a_{13}+a_{31}=0$, $2a_{14}=a_{51}$, $2a_{15}=a_{41}$, $a_{43}=-2a_{35}$, $a_{53}=-2a_{34}$, and $a_{44}=a_{55}$.

Case 8: Algebra $A_{1,2}\oplus A_{0,1}^3$.

Non-zero bracket: $[e_1,e_1]=e_2$.

Matrix entries: $a_{1i}=0$ for $i=2,3,4,5$.

Case 9: Algebra $A_{1,3}\oplus A_{0,1}^2$.

Non-zero brackets: $[e_1,e_1]=e_2$, $[e_3,e_3]=e_2$.

Coefficients: $a_{ij}=0$ for $i=1,3$ and $j=2,4,5$, and $a_{13}=a_{31}$.

Case 10: Algebra $A_{2,4}\oplus A_{0,1}$.

Non-zero brackets: $[e_1,e_1]=e_2$, $[e_3,e_4]=[e_4,e_3]=e_2$.

Matrix entries: $a_{ij}=a_{35}=0$ for $i=1,4$ and $j=2,5$, $a_{41}=a_{13}$, $a_{31}=a_{14}$, and $a_{33}=a_{44}$.

Case 11: Algebra $A_{1,2}^2\oplus A_{0,1}$.

Non-zero brackets: $[e_1,e_1]=e_2$, $[e_3,e_3]=e_4$.

Coefficients: $a_{ij}=0$ for $i=1,3$ and $j=2,4,5$, and $a_{13}=a_{31}$.

Case 12: Algebra $A_{1,2}\oplus A_{1,3}$.

Non-zero brackets: $[e_1,e_1]=e_2$, $[e_3,e_3]=e_4$, $[e_5,e_5]=e_4$.

Matrix entries: $a_{ij}=0$ for $i=1,3,5$ and $j=2,4$, and $a_{13}=a_{31}$.

Case 13: Algebra $A_{1,4}\oplus A_{0,1}$.

Non-zero brackets: $[e_1,e_1]=e_2$, $[e_1,e_3]=[e_3,e_1]=e_4$.

Matrix entries: $a_{1i}=a_{3j}=0$ for $i=2,3,4,5$ and $j=2,4,5$, and $a_{11}=a_{33}$.

For each algebra, the remaining matrix entries are free parameters, and the basis elements listed in Table 5 correspond to these free parameters after applying the matrix entries. The proof is completed. □

4. Discussion

4.1. Uniqueness of Matrix Representations

The uniqueness of our matrix representations requires careful discussion. The explicit matrix forms we provide are computed with respect to a given basis. Consequently, these representations are not canonical in an isomorphism-invariant sense, as they transform under conjugation by a change-of-basis matrix.

Nevertheless, these basis-dependent representations offer significant computational utility. They furnish explicit, verifiable forms directly applicable to calculations in mathematical physics and computational algebra. The individual matrix entries depend on the choice of basis, but the structural patterns we find (dimension of quasi-centroid, containment relation to centroid, characteristic block decompositions) are algebraic invariants.

Finally, our systematic approach guarantees reproducibility. Given the classification of the Mock-Lie algebra and its multiplication table, we can apply our method to obtain the corresponding quasi-centroid matrices in any preferred basis.

4.2. Comparing Centroids and Quasi-Centroids

Based on the known matrix representations of centroids [16], we can compare them with our new results of quasi-centroids for dimensions 2 to 5.

A general observation from this comparison is that the centroid is a proper subalgebra of the quasi-centroid of every Mock-Lie algebra we studied. This relation is directly derived from the definitions: the centroid condition $\phi \left(\right[x,y\left]\right)=\left[\phi \right(x),y]=[x,\phi (y\left)\right]$ implies a weaker quasi-centroid condition $\left[\phi \right(x),y]=[x,\phi (y\left)\right]$. The explicit matrices in Table 6, Table 7, Table 8 and Table 9 show that the quasi-centroid of low-dimensional Mock-Lie algebras has a larger structure than the centroid.

Table 6. Two-dimensional case.

Algebra	Centroid	Quasi-Centroid
$A_{1,2}$	$\left(\begin{array}{cc}{a}_{11}& 0\\ a_{21}& a_{11}\end{array}\right)$	$\left(\begin{array}{cc}{a}_{11}& 0\\ a_{21}& a_{22}\end{array}\right)$

Table 7. Three-dimensional cases.

Algebra	Centroid	Quasi-Centroid
$A_{1,2}\oplus A_{0,1}$	$\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{11}& a_{23}\\ a_{31}& 0& a_{33}\end{array}\right)$	$\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$
$A_{1,3}$	$\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{11}& a_{23}\\ 0& 0& a_{11}\end{array}\right)$	$\left(\begin{array}{ccc}{a}_{11}& 0& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{13}& 0& a_{33}\end{array}\right)$

Table 8. Four-dimensional cases.

Algebras	Centroid	Quasi-Centroid
$A_{1,2}\oplus A_{0,1}^2$	$\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}\\ a_{31}& 0& a_{33}& a_{34}\\ a_{41}& 0& a_{43}& a_{44}\end{array}\right)$	$\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{31}& a_{32}& a_{33}& a_{34}\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right)$
$A_{1,3}\oplus A_{0,1}$	$\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}\\ 0& 0& a_{11}& 0\\ a_{41}& 0& a_{43}& a_{44}\end{array}\right)$	$\left(\begin{array}{cccc}{a}_{11}& 0& a_{13}& 0\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{13}& 0& a_{33}& 0\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right)$
$A_{1,2}\oplus A_{1,2}$	$\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& 0\\ 0& 0& a_{33}& 0\\ a_{41}& 0& a_{43}& a_{33}\end{array}\right)$	$\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}\\ 0& 0& a_{33}& 0\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right)$
$A_{1,4}$	$\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& 0\\ a_{31}& 0& a_{11}& 0\\ a_{41}& a_{31}& a_{43}& a_{11}\end{array}\right)$	$\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{31}& 0& a_{11}& 0\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right)$
$A_{2,4}$	$\left(\begin{array}{cccc}{a}_{11}& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}\\ 0& 0& a_{11}& 0\\ 0& 0& 0& a_{11}\end{array}\right)$	$\left(\begin{array}{cccc}{a}_{11}& 0& a_{13}& a_{14}\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{14}& 0& a_{33}& a_{34}\\ a_{13}& 0& a_{43}& a_{33}\end{array}\right)$

Table 9. Five-dimensional cases.

Algebras	Centroid	Quasi-Centroid
$A_{1,5}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& 0& 0\\ 0& 0& a_{11}& 0& 0\\ a_{41}& 0& a_{43}& a_{11}& 0\\ a_{51}& 0& a_{53}& 0& a_{11}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ 0& 0& a_{11}& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$
$A_{2,5}$	$\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ a_{21}& 0& a_{23}& 0& 0\\ 0& 0& a_{33}& 0& 0\\ a_{41}& 0& 0& 0& 0\\ a_{51}& a_{52}& a_{53}& a_{33}& 0\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{31}& 0& a_{33}& a_{34}& a_{35}\\ a_{41}& 0& a_{31}& a_{11}& 0\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$
$A_{3,5}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}& 0\\ 0& 0& a_{11}& 0& 0\\ 0& 0& 0& a_{11}& 0\\ a_{51}& 0& a_{53}& a_{54}& a_{11}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& a_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{11}& 0& a_{33}& 0& 0\\ a_{41}& 0& a_{41}-a_{11}& a_{11}+a_{33}& 0\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$
$A_{4,5}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}& a_{25}\\ 0& 0& a_{11}& 0& 0\\ a_{41}& 0& a_{43}& a_{11}& a_{45}\\ 0& 0& 0& 0& a_{11}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ 0& 0& a_{33}& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ 0& 0& 0& 0& a_{55}\end{array}\right)$
$A_{5,5}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& 0& 0\\ 0& 0& a_{11}& 0& 0\\ 0& 0& 0& a_{11}& 0\\ 0& 0& 0& 0& a_{11}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ 0& 0& a_{33}& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ 0& 0& a_{53}& 0& a_{55}\end{array}\right)$
$A_{6,5}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}& 0\\ 0& 0& a_{11}& 0& 0\\ 0& 0& 0& a_{11}& 0\\ a_{51}& 0& a_{53}& a_{54}& a_{11}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{31}& 0& a_{11}& 0& 0\\ a_{41}& 0& 0& a_{44}& 0\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$
$A_{7,5}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}& a_{25}\\ 0& 0& a_{11}& 0& 0\\ 0& 0& 0& a_{11}& 0\\ 0& 0& 0& 0& a_{11}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& a_{13}& a_{14}& a_{15}\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ -a_{13}& 0& a_{33}& a_{34}& a_{35}\\ 2a_{15}& 0& -2a_{35}& a_{44}& a_{45}\\ 2a_{14}& 0& -2a_{34}& a_{54}& a_{44}\end{array}\right)$
$A_{1,2}\oplus A_{0,1}^3$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}& a_{25}\\ a_{31}& 0& a_{33}& a_{34}& a_{35}\\ a_{41}& 0& a_{43}& a_{44}& a_{45}\\ a_{51}& 0& a_{53}& a_{54}& a_{55}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{31}& a_{32}& a_{33}& a_{34}& a_{35}\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$
$A_{1,3}\oplus A_{0,1}^2$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}& a_{25}\\ 0& 0& a_{11}& 0& 0\\ a_{41}& 0& a_{43}& a_{44}& a_{45}\\ a_{51}& 0& a_{53}& a_{54}& a_{55}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& a_{13}& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{13}& 0& a_{33}& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$
$A_{2,4}\oplus A_{0,1}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& a_{24}& a_{25}\\ 0& 0& a_{11}& 0& 0\\ 0& 0& 0& a_{11}& 0\\ a_{51}& 0& a_{53}& a_{54}& a_{55}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& a_{13}& a_{14}& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{14}& 0& a_{33}& 0& 0\\ a_{13}& 0& a_{43}& a_{33}& 0\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$
$A_{1,2}^2\oplus A_{0,1}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& 0& a_{25}\\ 0& 0& a_{33}& 0& 0\\ a_{41}& 0& a_{43}& a_{33}& a_{45}\\ a_{51}& 0& a_{53}& 0& a_{55}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& a_{13}& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{13}& 0& a_{33}& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$
$A_{1,2}\oplus A_{1,3}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& 0& a_{25}\\ 0& 0& a_{33}& 0& 0\\ 0& 0& 0& a_{33}& 0\\ 0& 0& 0& 0& a_{33}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& a_{13}& 0& a_{15}\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{13}& 0& a_{33}& 0& a_{35}\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ a_{51}& 0& a_{53}& 0& a_{55}\end{array}\right)$
$A_{1,4}\oplus A_{0,1}$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{11}& a_{23}& 0& a_{25}\\ a_{31}& 0& a_{11}& 0& 0\\ a_{41}& a_{31}& a_{43}& a_{11}& a_{45}\\ a_{51}& 0& a_{53}& 0& a_{55}\end{array}\right)$	$\left(\begin{array}{ccccc}{a}_{11}& 0& 0& 0& 0\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{31}& 0& a_{11}& 0& 0\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\end{array}\right)$

4.3. Structural Analysis

Based on Table 6, Table 7, Table 8 and Table 9, we obtain three important patterns as follows:

• The centroid is always strictly contained within the quasi-centroid. This happens because the quasi-centroid condition $\left[\phi \right(x),y]=[x,\phi (y\left)\right]$ is less restrictive than the centroid condition $\phi \left(\right[x,y\left]\right)=\left[\phi \right(x),y]$. As a result, more linear maps can qualify.
• The quasi-centroid consistently has a higher dimension than the centroid. Relaxed constraints create additional degrees of freedom, which typically manifest as extra non-zero entries in the matrix representations.
• This strict containment clarifies the hierarchy between these structures. The centroid forms a proper subalgebra within the quasi-centroid. This reveals nested layers of symmetric structure in Mock-Lie algebras.

In summary, the quasi-centroid includes more symmetry-preserving linear maps than the centroid. This provides deeper insight into the symmetric nature of low-dimensional Mock-Lie algebras.

4.4. Applications in Mathematical Physics

The explicit matrix representations developed in this work enable concrete applications across multiple areas of mathematical and computational physics.

In molecular dynamics, particularly in quasi-centroid molecular dynamics (QCMD), our matrices offer a rigorous algebraic framework for describing symmetric transformations. A key finding is that the quasi-centroid includes more symmetry-preserving maps than the centroid. For instance, in the algebra $A_{1,3}$ (Theorem 2), the quasi-centroid contains the transformation $E_{13}+E_{31}$, which acts jointly on basis elements $e_1$ and $e_3$ while preserving the fundamental symmetry $\left[\phi \right(x),y]=[x,\phi (y\left)\right]$. Such additional mappings can model complex physical interactions—such as vibrational couplings or external field effects—that comply with symmetry but lie beyond the scope of the centroid. Our classification thus helps identify and design symmetry-respecting operators for molecular simulations [18].

In symmetry analysis, the quasi-centroid condition $\left[\phi \right(x),y]=[x,\phi (y\left)\right]$ states a basic symmetry constraint for physical systems. For example, in quantum angular momentum algebras, it helps find transformations that maintain rotational symmetry. Our low-dimensional matrix models make it easier to detect such symmetry-preserving deformations [23].

From a computational perspective, our classification can be directly used in symbolic algebra systems. This supports automated symmetry analysis in broad studies of Lie algebras and deformation theory [21].

These applications show the practical utility of our theoretical results beyond classification.

4.5. Concluding Remarks

This paper gives a complete classification of quasi-centroids, clarifying the structure of linear symmetries in low-dimensional Mock-Lie algebras. Our explicit matrix representations provide a solid computational basis for studying transformations that preserve the symmetry $\left[\phi \right(x),y]=[x,\phi (y\left)\right]$. These results contribute to both theoretical understanding and practical tools for research on algebraic structures and their physical applications.